Greenberg’s conjecture

Greenberg’s conjecture predicts that, for a totally real number field $k$ and prime $p$, the unramified Iwasawa module $X(k_\infty^{\mathrm c})$ is finite. The general assertion remains unresolved.

Known results

• February 2022: Greenberg’s conjecture was proved for cyclotomic $\mathbb{Z}_2$-extensions of real quadratic fields $\mathbb{Q}(\sqrt f)$ with $0<f<10000$.
• April 2024: It was verified for an infinite family $\mathbb{Q}(\sqrt{pqr})$ under explicit hypotheses, with $\lambda=0$.
• May 2025: Nguyen Quang Do obtained pseudo-nullity criteria for generalized Greenberg conjectures in families of number fields.
• May 2026: $\lambda_2(K)=0$ was proved for a specified family $K=\mathbb{Q}(\sqrt{pq})$.

September 2026 restricted-ramification criteria

On September 22, 2026, Tsuyoshi Itoh and Naoki Kumakawa’s preprint developed a restricted-$p$-ramification method and gave necessary-and-sufficient cyclicity criteria in a quadratic-field setting. This narrows the conjecture in explicit families but does not prove it in general.

Current status (as of September 2026): Greenberg’s conjecture is verified for several explicit families, while the general case remains open; the latest preprint claims further conditional progress, not a general solution.